Abstract We explore whether quantum advantages can be found for the zeroth-order online convex optimization (OCO) problem, which is also known as bandit convex optimization with multi-point feedback. In this setting, given access to zeroth-order oracles (that is, the loss function is accessed as a black box that returns the function value for any queried input), a player attempts to minimize a sequence of adversarially generated convex loss functions. This procedure can be described as a T round iterative game between the player and the adversary. In this paper, we present quantum algorithms for the problem and show for the first time that potential quantum advantages are possible for problems of OCO. Specifically, our contributions are as follows. (i) When the player is allowed to query zeroth-order oracles O (1) times in each round as feedback, we give a quantum algorithm that achieves O ( T ) regret without additional dependence of the dimension n , which outperforms the already known optimal classical algorithm only achieving O ( n T ) regret. Note that the regret of our quantum algorithm has achieved the lower bound of classical first-order methods. (ii) We show that for strongly convex loss functions, the quantum algorithm can achieve O (log T ) regret with O (1) queries as well, which means that the quantum algorithm can achieve the same regret bound as the classical algorithms in the full information setting.
Abstract We explore whether quantum advantages can be found for the zeroth-order online convex optimization (OCO) problem, which is also known as bandit convex optimization with multi-point feedback. In this setting, given access to zeroth-order oracles (that is, the loss function is accessed as a black box that returns the function value for any queried input), a player attempts to minimize a sequence of adversarially generated convex loss functions. This procedure can be described as a T round iterative game between the player and the adversary. In this paper, we present quantum algorithms for the problem and show for the first time that potential quantum advantages are possible for problems of OCO. Specifically, our contributions are as follows. (i) When the player is allowed to query zeroth-order oracles O (1) times in each round as feedback, we give a quantum algorithm that achieves O ( T ) regret without additional dependence of the dimension n , which outperforms the already known optimal classical algorithm only achieving O ( n T ) regret. Note that the regret of our quantum algorithm has achieved the lower bound of classical first-order methods. (ii) We show that for strongly convex loss functions, the quantum algorithm can achieve O (log T ) regret with O (1) queries as well, which means that the quantum algorithm can achieve the same regret bound as the classical algorithms in the full information setting.
As a fundamental tool in AI, convex optimization has been a significant research field for many years, and the same goes for its online version. Recently, general convex optimization problem has been accelerated with the help of quantum computing, and the technique of online convex optimization has been used for accelerating the online quantum state learning problem, thus we want to study whether online convex optimization (OCO) model can also be benefited from quantum computing. In this paper, we consider the OCO model, which can be described as a T round iterative game between the player and the adversary. A key factor for measuring the performance of an OCO algorithm A is the regret denoted by regretT(A), and it is said to perform well if its regret is sublinear as a function of T. Another factor is the computational cost (e.g., query complexity) of the algorithm. We give a quantum algorithm for the online convex optimization model with only zeroth-order oracle available, which can achieve O(T) and O(logT) regret for general convex loss functions and α-strong loss functions respectively, where only O(1) queries are needed in each round. Our results show that the zeroth-order quantum oracle is as powerful as the classical first-order oracle, and show potential advantages of quantum computing over classical computing in the OCO model where only zeroth-order oracle available.
We explore whether quantum advantages can be found for the zeroth-order feedback online exp-concave optimization problem, which is also known as bandit exp-concave optimization with multi-point feedback. We present quantum online quasi-Newton methods to tackle the problem and show that there exists quantum advantages for such problems. Our method approximates the Hessian by quantum estimated inexact gradient and can achieve O(nlogT) regret with O(1) queries at each round, where n is the dimension of the decision set and T is the total decision rounds. Such regret improves the optimal classical algorithm by a factor of T2/3.
We initiate the study of quantum algorithms for optimizing approximately convex functions. Given a convex set K⊆Rn and a function F:Rn→R such that there exists a convex function f:K→R satisfying supx∈K∣F(x)−f(x)∣≤ε/n, our quantum algorithm finds an x∗∈K such that F(x∗)−minx∈KF(x)≤ε using O~(n3) quantum evaluation queries to F. This achieves a polynomial quantum speedup compared to the best-known classical algorithms. As an application, we give a quantum algorithm for zeroth-order stochastic convex bandits with O~(n5log2T) regret, an exponential speedup in T compared to the classical Ω(T) lower bound. Technically, we achieve quantum speedup in n by exploiting a quantum framework of simulated annealing and adopting a quantum version of the hit-and-run walk. Our speedup in T for zeroth-order stochastic convex bandits is due to a quadratic quantum speedup in multiplicative error of mean estimation.
We study non-linear bandit optimization where the learner maximizes a black-box function with zeroth order function oracle, which has been successfully applied in many critical applications such as drug discovery and materials design. Existing works have showed that with the aid of quantum computing, it is possible to break the classical Ω(√T) regret lower bound and achieve the new O(poly log T) upper bound. However, they usually assume that the objective function sits within the reproducing kernel Hilbert space and their algorithms suffer from the curse of dimensionality. In this paper, we propose the new Q-NLB-UCB algorithm which enjoys an input dimension-free O(poly log T) upper bound, making it applicable for high-dimensional tasks. At the heart of our algorithm design are quantum Monte Carlo mean estimator, parametric function approximation technique, and a new quantum non-linear regression oracle, which can be of independent interests in more quantum machine learning problems. Our algorithm is also validated for its efficiency compared with other quantum algorithms on both high-dimensional synthetic and real-world tasks.
We study non-linear bandit optimization where the learner maximizes a black-box function with zeroth order function oracle, which has been successfully applied in many critical applications such as drug discovery and materials design. Existing works have showed that with the aid of quantum computing, it is possible to break the classical Ω(T) regret lower bound and achieve the new O(polylogT) upper bound. However, they usually assume that the objective function sits within the reproducing kernel Hilbert space and their algorithms suffer from the curse of dimensionality. In this paper, we propose the new Q-NLB-UCB algorithm which enjoys an \emph{input dimension-free} O(polylogT) upper bound, making it applicable for high-dimensional tasks. At the heart of our algorithm design are quantum Monte Carlo mean estimator, parametric function approximation technique, and a new quantum non-linear regression oracle, which can be of independent interests in more quantum machine learning problems. Our algorithm is also validated for its efficiency compared with other quantum algorithms on both high-dimensional synthetic and real-world tasks.
This work investigates the possibility of quantum speedups for continuous optimization through quantum Hamiltonian simulation. We establish the first rigorous query complexity bounds for unconstrained convex optimization via digital quantum annealing, based on the non-adiabatic Quantum Hamiltonian Descent (QHD) framework. In the process, we derive rigorous resource estimates for digital quantum simulation of Schrödinger operators depending only on input simulation parameters, given black-box evaluation access to separable, Lipschitz continuous potential b(t) f(x). We apply these simulation bounds to assess the complexity of high-dimensional convex optimization. Our annealing schedule achieves arbitrarily fast convergence rates in the evolution time, with computational time determined solely by the cost of discretization. We show that a G-Lipschitz convex function can be optimized to an error of ε with 𝒪̃(d^{1.5} G² R²/ε²) queries, given a starting point that is Euclidean distance R from optimal. Under reasonable assumptions such as the complexity of simulating Schrödinger operators, we show that Ω̃(d/ε²) queries are necessary. This suggests QHD does not offer improvements over classical methods in the noiseless zeroth order setting. However, we show that the QHD algorithm can tolerate 𝒪̃(ε³ /d^{1.5} G² R²) noise in function evaluation, and as a result, provides a super-quadratic query advantage over the best existing noise-tolerant classical algorithms in the high-dimensional setting. We leverage this to design a quantum algorithm for stochastic convex optimization that offers a super-quadratic speedup over all known classical and quantum algorithms in this regime. To our knowledge, these results represent the first rigorous quantum speedups for convex optimization obtained through a dynamical algorithm.